Geodesic
Free products of two real cyclic matrix groups
Glasgow mathematical journal, Tome 15 (1974) no. 2, pp. 121-128

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We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products . 
Evans, R. J. Free products of two real cyclic matrix groups. Glasgow mathematical journal, Tome 15 (1974) no. 2, pp. 121-128. doi: 10.1017/S0017089500002287
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